Architectural Origami
Architectural Form Design Systems based on Computational Origami
Tomohiro Tachi Graduate School of Arts and Sciences, The University of Tokyo JST PRESTO
1 0
Introduction
2 Background 1: Origami
Running Hare 2008 Tetrapod 2009 Origami Teapot 2007 Tomohiro Tachi Tomohiro Tachi Tomohiro Tachi 3 Background 2: Applied Origami • Static:
Photograph of origami dome removed – Manufacturing due to copyright restrictions. • Forming a sheet • No Cut / No Stretch • No assembly – Structural Stiffness Table (T. Tachi and D. Koschitz) • Dynamic: – Deployable structure Photograph of solar panels removed • Mechanism due to copyright restrictions. • Packaging – Elastic Plastic Property • Textured Material • Energy Absorption • Continuous surface
Potentially useful for – Adaptive Environment – Context Customized Design – Personal Design
– Fabrication Oriented Design 4 Architectural Origami Pattern - 2D Pattern
• Origami Architecture Direct application of Origami for Design - Static Shape – Design is highly restricted by the symmetry of the original pattern – Freeform design results in losing important property (origami-inspired design) Design Apply • Architectural Origami Origami theory for Design Conditions – Extract characteristics of origami - 3D shape in motion – Obtain solution space of forms from - Behavior the required condition and design context
5 Outline
1. Origamizer – tucking molecules – layout algorithm 2. Freeform Origami – constraints of origami – perturbation based calculation – mesh modification 3. Rigid Origami – simulation – design by triangular mesh – design by quad mesh – non-disk? 6 1 Origamizer Related Papers: •Demaine, E. and Tachi, T. “Origamizer: A Practical Algorithm for Folding Any Polyhedron,” work in progress.
•Tachi, T.,“Origamizing polyhedral surfaces,” IEEE Transactions on Visualization and Computer Graphics, vol. 16, no. 2, 2010.
•Tachi, T., “Origamizing 3d surface by symmetry constraints,” August 2007. ACM SIGGRAPH 2007 Posters.
•Tachi, T., “3D Origami Design based on Tucking Molecule,” in Origami4: A K Peters Ltd., pp. 259-272, 2009. 7 Existing Origami Design Method by Circle Packing
Diagram of circle river method removed due to copyright restrictions.
Photograph and crease pattern for Scutigera removed due to copyright restrictions.
8 1D vs. 3D
• Circle River Method / Tree Method – Works fine for tree-like objects – Does not fit to 3D objects
• Origamizer / Freeform Origami – 3D Polyhedron, surface approximation – What You See Is What You Fold
9 3D Origami
Laptop PC 2003 by Tomohiro Tachi not completed
10 3D Origami
Human 2004
Tetrapod 2009 Tomohiro Tachi
11 3D Origami Roofs 2003
Everything seems to be possible! 12 Problem: realize arbitrary polyhedral surface with a developable surface
• Geometric Constraints Input: – Developable Surf Arbitrary Polyhedron – Piecewise Linear – Forget about Continuous Folding Motion Output: Crease Pattern • Potential Application – Fabrication by folding and bending
Folded Polyhedron
13 Approach: Make “Tuck”
• Tuck develops into – a plane • Tuck folds into – a flat state hidden behind polyhedral surface →Important Advantage: We can make Negative Curvature Vertex 14 Basic Idea
Origamize Problem ↓ Lay-outing Surface Polygons Properly ↓ Polygons Tessellating Surface Edge-Tucking Polygons and “Tucking Molecule Vertex-Tucking Molecules” Molecule ↓ Parameter everything by Tucking Molecule: – Angle θ (i, j) – Distance w (i, j) θ (j, i) = −θ (i, j) w(j, i) = w(i, j) + 2λ(i, j) sin(0.5θ (i, j)) 15 Geometric Constraints (Equations)
N −1 N −1 ∑ ()()ji n −= ∑απθ ,2, ji n )1( n=0 n=0 n N −1 cos()Θm 0 ∑m=1 ∑ (), jiw n n = )2( n=0 sin Θ 0 ()∑m=1 m 1 1 where =Θ θ ()()− α ,, ++ θ (), jijiji m 2 m 1 m 2 m Two-Step Linear Mapping 1. Mapping based on (1) (linear)
w = bwC 2. Mapping based on (2) (linear) + + + −+= wCCIbCw where Cw is the generalized inverse of Cw ( Nw edge ww ) 0 + T T −1 m atrix theIf matrix is rank,-full = ()CCCC wwww
gives (Nedge-2Nvert)dimensional solution space (within the space, we solve the inequalities)
16 Geometric Constraints (Inequalities)
• 2D Cond. (),oi ≥ πθ – Convex Paper ()oiw ≥ 0, Convexity of paper Non intersection (convexity of molecule) – Non-intersection (), ji <<− πθπ min( ()()ijwjiw ) ≥ 0,,,
0 m <Θ≤ π – Crease pattern non-intersection 1 ()cos,2 θ (), jiji φ(), ji ≤ arctan 2 + 5.0 π ()()+ ,, ijwjiw • 3D Cond. 展開図の妥当条件： 頂点襞分子iと稜線襞分子ijが共有す for tuck proxy angle τ ′()and, ji depth ′()id る辺を含むDelaunay三角形の頂点角φ(i, j)を用いる。 – Tuck angle condition 1 (), ()(), −≤− τπθφ ′ , jijiji 2 – Tuck depth condition 1 ()()jiw ≤ 2, sin ′ , − ατ ()(), ′ idjiji 2
17 Design System: Origamizer
3D CP • Auto Generation of Crease Pattern • Interactive Editing (Search within the solution space) – Dragging Developed Facets
– Edge Split
– Boundary Editing
Developed in the project “3D Origami Design Tool”
of IPA ESPer Project 18 19 How to Fold Origami Bunny
0. Get a crease pattern using Origamizer 20 1. Fold Along the Crease Pattern 21 22 2. Done! 23 Proof? Ongoing joint work with Erik Demaine
24 2
Freeform Origami
Related Papers: •Tomohiro Tachi, "Freeform Variations of Origami", in Proceedings of The 14th International Conference on Geometry and Graphics (ICGG 2010), Kyoto, Japan, pp. 273-274, August 5-9, 2010. (to appear in Journal for Geometry and Graphics Vol. 14, No. 2)
•Tomohiro Tachi: "Smooth Origami Animation by Crease Line Adjustment ," ACM SIGGRAPH 2006 Posters, 2006. 25 Objective of the Study
1. freeform – Controlled 3D form – Fit function, design context, preference, ...
2. origami utilize the properties – Developability → Manufacturing from a sheet material based on Folding, Bending – Flat-foldability → Folding into a compact configuration or Deployment from 2D to 3D – Rigid-foldability → Transformable Structure – Elastic Properties
... 26 Proposing Approach
• Initial State: existing origami models (e.g. Miura- ori, Ron Resch Pattern, …) + Perturbation consistent with the origami conditions. • Straightforward user interface.
27 Model
• Triangular Mesh (triangulate quads) x1 • Vertex coordinates represent the configuration y1 – 3N variables, where N is the # of z1 v v vertices X = x Nv yNv • The configuration is zNv constrained by developability, flat-foldability, …
28 Developability
Engineering Interpretation → Manufacturing from a sheet material based on Folding, Bending
• Global condition – There exists an isometric map to a plane. (if topological disk) • Local condition ⇔– Every point satisfies • Gauss curvature = 0
29 Developable Surface
• Smooth Developable Surface – G2 surface (curvature continuous) • "Developable Surface" (in a narrow sense) • Plane, Cylinder, Cone, Tangent surface – G1 Surface (smooth, tangent continuous) pl ane cyl i nder cone t angent • "Uncreased flat surface" • piecewise Plane, Cylinder, Cone, Tangent surface
• Origami – G0 Surface – piecewise G1 Developable G0 Surface
30 Developability condition to be used • Constraints – For every interior vertex v (kv-degree), gauss area equals 0. GC<0 GC=0 GC>0
kv Gv 2 ∑θπ i =−= 0 i=0
31 Flat-foldability
Engineering Interpretation → Folding into a compact configuration or Deployment from 2D to 3D
• Isometry condition • isometric mapping with mirror reflection • Layering condition • valid overlapping ordering • globally : NP Complete [Bern and Hayes 1996]
32 Flat-foldability condition to be used
– Isometry Alternating sum of angles is 0 [Kawasaki 1989] ⇔ kv Fv = ∑ ()i θi = 0sgn i=0 – Layering [kawasaki 1989]
– If θi is between foldlines assigned with MM ⇒ or VV, θi min(θi−1,θi+1) + empirical condition [tachi 2007] ≧ – If θi and θi+1 are composed byfoldlines assigned with MMV or VVM then, θi θi+1 ≧ 33 Other Conditions
• Conditions for fold angles – Fold angles ρ – V fold: 0<ρ<π – M fold: −π<ρ<0 – crease: −απ<ρ<απ (α=0:planar polygon) • Optional Conditions – Fixed Boundary • Folded from a specific shape of paper – Rigid bars – Pinning
34 Settings
x1 • Initial Figure: y – Symmetric Pattern 1 z1 • Freeform Deformation X = – Variables (3Nv) • Coordinates x X Nv – Constraints (2Nv_in+Nc) yNv • Developability zNv • Flat-foldability • Other Constraints Under-determined System
→Multi-dimensional Solution Space
35 Solve Non-linear Equation
The infinitesimal motion satisfies: ∂G ∂G ∂G ∂ρ ∂θ ∂X ∂θ kv ∂θ 1 ∂ ∂ ∂ ijk −= bT F F F ∂X G 2 ∑θπ =−= 0 ij XC = X = = 0X v i ∂x iji ∂X ∂θ ∂ρ ∂ρ i=0 kv ∂θ 11 ∂H ∂H ∂ ijk T += bb T H ∂X Fv = ∑ ()i θi = 0sgn ij jk ∂x ijj jk ∂X ∂θ ∂ρ i=0 ∂θijk 1 T For an arbitrarily given (through GUI) −= b jk ∂x jkk Infinitesimal Deformation ∆X0
Euler Integration
+ ∆− XCC 0
+ ∆ − rC + + X0 ()3N ∆−+−=∆ XCCIrCX 0 ∆X v = rr Calculated Trajectory
= 0r Ideal Trajectory 36 Freeform Origami
3D Get A Valid Value • Iterative method to calculate the conditions Developed • Form finding through User Interface Implementation • Lang Flat-folded – C++, STL • Library – BLAS (intel MKL) • Interface – wxWidgets, OpenGL To be available on web
37 Mesh Modification Edge Collapse • Edge Collapse [Hoppe etal 1993]
• Maekawa’s Theorem [1983] for flat foldable pattern M - V = ±2
38 Mesh Modification
39 Miura-Ori
• Original – [Miura 1970] • Application – bidirectionally expansible (one- DOF) Photograph of solar panels removed due to copyright restrictions. – compact packaging – sandwich panel • Conditions – Developable – Flat-foldable – op: (Planar quads)(→Rigid Foldable)
40 Miura-ori Generalized
• Freeform Miura-ori
41 Miura-ori Generalized
Origami Metamorphose Tomohiro Tachi 2010 42 Ron Resch Pattern
Photograph of origami model removed • Original due to copyright restrictions. – Resch [1970] • Characteristics – Flexible (multiDOF) – Forms a smooth flat surface + scaffold
• Conditions – Developable – 3-vertex coincidence
43 Ron Resch Pattern Generalized
44 Generalized Ron Resch Pattern
45 Crumpled Paper
• Origami = crumpled paper = buckled sheet
• Conditions – Developable – Fixed Perimeter
46 crumpled paper example
47 48 Waterbomb Pattern
Screenshot from video removed due to copyright restrictions. To view video: https://www.youtube.com/watch?v=SvDSNDR0oXo. • “Namako” (by Shuzo Fujimoto) • Characteristics – Flat-foldable – Flexible(multi DOF) – Complicated motion
• Application Photograph of metal origami heart stent removed due to copyright restrictions. – packaging – textured material – cloth folding...
49 Waterbomb Pattern Generalized
50 51 52 3
Rigid Origami
•Tachi T.: "Rigid-Foldable Thick Origami", in Origami5, to appear. •Tachi T.: "Freeform Rigid-Foldable Structure using Bidirectionally Flat-Foldable Planar Quadrilateral Mesh", Advances in Architectural Geometry 2010, pp. 87--102, 2010. •Miura K. and Tachi T.: "Synthesis of Rigid-Foldable Cylindrical Polyhedra," Journal of ISIS- Symmetry, Special Issues for the Festival-Congress Gmuend, Austria, August 23-28, pp. 204-213, 2010. •Tachi T.: "One-DOF Cylindrical Deployable Structures with Rigid Quadrilateral Panels," in Proceedings of the IASS Symposium 2009, pp. 2295-2306, Valencia, Spain, September 28- October 2, 2009. •Tachi T.: "Generalization of Rigid-Foldable Quadrilateral-Mesh Origami," Journal of the International Association for Shell and Spatial Structures (IASS), 50(3), pp. 173–179, December 2009. •Tachi T.: "Simulation of Rigid Origami ," in Origami4, pp. 175-187, 2009.
53 Rigid Origami?
• Rigid Origami is – Plates and Hinges model for origami • Characteristics – Panels do not deform • Do not use Elasticity – synchronized motion • Especially nice if One-DOF – watertight cover for a space • Applicable for – self deployable micro mechanism – large scale objects under gravity using thick panels
54 Study Objectives
1. Generalize rigid foldable structures to freeform 1. Generic triangular-mesh based design • multi-DOF • statically determinate 2. Singular quadrilateral-mesh based design • one-DOF • redundant contraints 2. Generalize rigid foldable structures to cylinders and more
55 Examples of Rigid Origami
56 Basics of Rigid Origami Angular Representation C2(ρ2) C3(ρ3) χ = BC 1211 • Constraints B23 λ2 λ3 B12 χ = BC C1(ρ1) 2322 – [Kawasaki 87] B34 λ1 B41 χ = BC 3433 [belcastro and Hull 02] λ4 χ = BC 4144
1 −1χχχ nn = I C4(ρ4)
– 3 equations per interior Generic case: vertex DOF = Ein -3Vin • Vin interior vert + + Ein foldline model: ρ []N −= CCI ρ 0 – constraints: = + []C ρ 0 where C is the × 3 EV inin matrix -pseudo inverse of C 57 DOF in Generic Triangular Mesh
Euler’s:(Vin+Eout)-(Eout+Ein)+F=1
Triangle : 3F=2Eout + Ein
Mechanism: DOF = Ein -3Vin
Disk with Eout outer edges
DOF = Eout -3 with H generic holes
DOF = Eout -3 -3H
(Vin+Eout)-(Eout+Ein)+F=1-H
DOF = Ein -3Vin-6H
58 Hexagonal Tripod Shell
Hexagonal boundary:
Eout = 6 DOF = 6 - 3 =3 + rigid DOF = 6 ∴
3 pin joints (x,y,z): 3x3 = 9 constraints
∴
59 60 Generalize Rigid-Foldable Planar Quad-Mesh
• One-DOF – Every vertex transforms in the same way – Controllable with single actuator • Redundant – Rigid Origami in General • DOF = N – 3M • N: num of foldlines N=4, M=1 • M: num of inner verts DOF = 1 – nxn array N=2n(n-1), M=(n-1)2 -> DOF=-(n-2)2+1 -> n>2, then overconstrained if not singular – Rank of Constraint Matrix is N-1 • Singular Constraints – Robust structure – Improved Designability 61 Idea: Generalize Regular pattern
• Original – Miura-ori – Eggbox pattern
• Generalization To: Generalize Non Symmetric forms
(Do not break rigid foldability)
62 Flat-Foldable Quadrivalent Origami MiuraOri Vertex • one-DOF structure – x,y in the same direction
• Miura-ori • Variation of Miura-ori
63 Flat-Foldable Quadrivalent Origami MiuraOri Vertex • Intrinsic Measure:
0 −= θπθ 2
1 −= θπθ 3 • Folding Motion – Opposite fold angles are equal −= ρρ – Two pairs of folding motions 31
are linearly related. = ρρ 20 ρ 1 cos()−+ θθ ρ tan 0 = 10 tan 1 2 1 cos()++ θθ 10 2
64 Flat-Foldable Quadrivalent Origami MiuraOri Vertex
ρ1()t ρ ()t01 tan tan 2 2 −= ρρ ρ ()t ρ ()t 31 2 02 tan = λ()t tan 2 2 = ρρ 20 ρ N ()t ρ N ()t0 ρ 1 cos()−+ θθ ρ tan tan 0 = 10 1 2 2 tan tan 2 1 cos()++ θθ 10 2
65 Get One State and Get Continuous Transformation Finite Foldability: Existence of Folding Motion There is one static state with ⇔ • Developability • Flat-foldability • Planarity of Panels
ρ1()t ρ ()t01 tan tan 2 2 ρ2 ()t ρ ()t02 tan = λ tan 2 ()t 2 ρ N ()t ρ N ()t0 tan tan 2 2
66 Built Design
• Material – 10mm Structural Cardboard (double wall) – Cloth • Size – 2.5m x 2.5m
• exhibited at NTT ICC
67 68 Rigid Foldable Curved Folding
• Curved folding is rationalized by Planar Quad Mesh • Rigid Foldable Curved Folding = Curved folding without ruling sliding
69 70 Discrete Voss Surface Eggbox-Vertex • one-DOF structure – Bidirectionally Flat-Foldable
• Eggbox-Pattern • Variation of Eggbox Pattern
71 Discrete Voss Surface Eggbox-Vertex • Intrinsic Measure:
= θθ 20
= θθ 31 Complementary Folding Angle • Folding Motion = ρρ −= ′ −= ρπρπ ′ – Opposite fold angles are 31 1 3 equal = ρρ 20 – Two pairs of folding motions ρ 1 cos()−+ θθ ρ tan 0 = 10 cot 1 are linearly related. 2 1 cos()++ θθ 2 [SCHIEF et.al. 2007] 10 1 cos()−+ θθ ρ′ = 10 tan 1 1 cos()++ θθ 10 2
72 Eggbox: Discrete Voss Surface
• Use Complementary Folding Angle for “Complementary Foldline” Complementary Folding Angle ′ ′ = ρρ 31 −= 1 −= ρπρπ 3 ρ ()t ρ ()t tan 0 tan 00 = ρρ 2 2 20 ρ ()t ρ ()t 1 01 ρ0 1 cos()−+ θθ 10 ρ1 tan = λ()t tan tan = cot 2 2 2 1 cos()++ θθ 2 10 ρ N ()t ρ N ()t0 ′ tan tan 1 cos()−+ θθ 10 ρ1 2 2 = tan 1 cos()++ θθ 10 2
73 Hybrid Surface: BiDirectionally Flat-Foldable PQ Mesh • use 4 types of foldlines – mountain fold • 0° -> -180° – valley fold • 0° -> 180°
– complementary 3D mountain fold • -180°-> 0° – complementary Developed valley fold • 180° -> 0° “developed” flat-folded
state state Flat-folded 74 Developability and Flat-Foldability
• Developed State: – Every edge has fold angle complementary fold angle to be 0° 3 dev ∑ ()i θσ i = 40 + 22 CFForCF i=0 3 2 ∑θπ i =− 0 4F • Flat-folded State: i=0 – Every edge has fold angle complementary fold angle to 3 be ±180° ff ∑ ()i θσ i = + 2240 CFForF i=0 3 2 ∑θπ i =− 0 4CF = i 0 75 Hybrid Surface: BiDirectionally Flat-Foldable PQ Mesh
76 77 3b
Rigid-foldable Cylindrical Structure
78 Topologically Extend Rigid Origami
• Generalize to the cylindrical, or higher genus rigid- foldable polyhedron.
• But it is not trivial!
79 Rigid-Foldable Tube Basics Miura-Ori Reflection
(Partial Structure of Thoki Yenn’s “Flip Flop”)
80 Symmetric Structure Variations
81 Parametric design of cylinders and composite structures
82 Cylinder -> Cellular Structure [Miura & Tachi 2010]
83 84 85 Isotropic Rigid Foldable Tube Generalization • Rigid Foldable Tube based on symmetry
• Based on – “Fold” – “Elbow”
= special case of BDFFPQ Mesh
86 Generalized Rigid Folding Constraints
• For any closed loop in Mesh
TTT kkk −−− 0,11,21,0 = I
where Ti,j is a 4x4 transformation matrix to translate facets coordinate i to j • When it is around a vertex: T is a rotation matrix.
87 Generalized Rigid Folding Constraints
• If the loop surrounds no hole: – constraints around each vertex • If there is a hole, – constraints around each vertex + 1 Loop Condition
88 Loop Condition : Sufficient Condition loop condition for finite rigid foldability
→ Sufficient Condition : start from symmetric cylinder and fix 1 loop
89 90 Manufactured From Two Sheets of Paper
91 92 93 94 95 96 97 98 99 Thickening
• Rigid origami is ideal surface (no thickness) • Reality: – There is thickness – To make “rigid” panels, thickness must be solved geometrically • Modified Model: – Thick plates – Rotating hinges at the edges
100 Hinge Shift Approach
• Main Problem – non-concurrent edges →6 constraints (overconstrained) • Symmetric Vertex: – [Hoberman 88] – use two levels of thickness – works only if the vertex is symmetric (a = b, c=d=π-a) • Slidable Hinges – [Trautz and Kunstler 09] – Add extra freedom by allowing „slide“ – Problem: global accumulation of slide (not locally designable)
101 Our Approach
Hinge Shift Volume Trim
Non-concurrent edges Concurrent edges
102 Trimming Volume
• folds up to −δπ • offsetting edges by δ t cot →Different speed 2 for each edge: Weighed Straight Skeleton
103 Variations
• Use constant thickness panels – if both layers overlap sufficiently • use angle limitation – useful for defining the “deployed 3D state”
104 Example
105 Example
• Constant Thickness Model – the shape is locally defined – cf: Slidable Hinge →
106 厚板のカッティングパターン生成
• 実装 – Grasshopper + C# components (Rhinoceros Plugin)
• →二次元パターン – 2軸CNCマシンで構築可能 • カッティングプロッタ • レーザーカッター • ウッドルータ – 組み立ての合理化の可能性
107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 Example: Construct a foldable structure that temporarily connects existing buildings • Space: Flexible – Connects when opened • Openings: different position and orientation • Connected gallery space – Compactly folded • to fit the facade • Structure: Rigid – Rigid panels and hinges
122 Panel Layout
123 124 125 126 127 128 129 MIT OpenCourseWare http://ocw.mit.edu
6.849 Geometric Folding Algorithms: Linkages, Origami, Polyhedra Fall 2012
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